Biomedical Engineering Reference
In-Depth Information
or including a more detailed representation for
I
m
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
−
2
R
2
R
...
V(
1
)
V(
2
)
V(
3
)
.
V(n
−
1
R
−
2
R
1
R
...
−
2
R
1
R
1
R
0
...
.
.
.
.
.
.
1
R
−
2
R
1
R
2
)
...
0
1
R
−
2
R
1
R
V(n
−
1
)
...
V (n)
2
R
−
2
R
...
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
V(
1
)
V(
2
)
V(
3
)
.
V(n
I
ion
(
1
)
I
ion
(
2
)
I
ion
(
3
)
.
I
ion
(n
d
dt
=
C
m
+
−
2
)
−
2
)
V(n
−
1
)
I
ion
(n
1
)
I
ion
(n)
−
V (n)
or more compactly,
d
V
dt
+
AV
=
C
m
I
ion
(5.16)
where the bold text indicates a vector or matrix.
A
is called a
coupling
matrix because the location of
the entries show exactly how nodes are connected. Each row of
A
corresponds to one node. Although
we have assumed
R
is constant, in reality we know that the resistance will depend upon
a
and
dx
.In
principle,
R
could vary in the matrix
A
and will indicate the strength of the connections.
To solve Eq. (5.16) we can perform the same rearrangement as in Sec. 2.5
C
m
AV
I
ion
dt
d
V
=
−
(5.17)
V
new
=
V
old
+
d
V
.
(5.18)
Besides being a compact way of writing the simultaneous equations of a cable, there is a practical
reason for the
vectorized
form of Eq. (5.17). There exist many sophisticated methods of multiplying,
adding, and factoring vectors and matrices that can drastically speedup the simulations.
Homework Problems
(1) Create the
A
coupling matrix for a cable where
R
i
=
100
cm
.
(2) How does the form of the matrix derived in problem 1 change if the cable branches?
